Is It Faster to Fall Through the Earth or Orbit to the Other Side? An Example of Gauss’s Law for Gravity

Image: Screen capture from VPython showing the Earth with a tunnel

How could you make a tunnel through the center of the Earth? Well, you probably couldn’t. However — if you COULD, it would make a great physics problem. It goes like this.

Suppose the Earth has a uniform density (that part is important) and a hole that goes right through the center. You can drop an object in the hole and it will fall from the gravitational force. Oh, don’t worry about air resistance — all the air has been pumped out of the hole (if we can dig through the Earth, surely we could get the air out). Question: how long would it take the object to fall to the other side of the Earth? Would it be faster to orbit to get to the other side instead?

Yeah, there’s a bunch of stuff in that question. That’s what makes it so much fun. Let’s get started.

Gauss’s Law and the Gravitational Field

If you have a point mass (yes, like the Earth), we can calculate the gravitational field (g) as:

Here G is the universal gravitational constant (6.67 x 10^-11 N*m²/kg²) and M is the mass of the Earth (5.972 x 10²⁴ kg). The vector r is from the center of the planet to some location. Note that r-hat is a unit vector to make the gravitational field a…